Write 2 log3 x + log3 5 as a single logarithmic expression.
You will need to make use of the following rules of logarithm:
1. a.logb(c) = logb(c^a)
2. logb(a) + logb(c) = logb(a.c)
2.log3 (x) + log3 (5)
= log3 (x^2) + log3 (5)
= log3 (5x^2)
Just a warning:
Look at rule number 2 above.
It is logb(a) + logb(c) = logb(a.c)
Never end up remembering " logb(a).logb(c) = logb(a+c) " !! There is no such rule at all!
I have a little memory aid to share:
"I remember that logarithm is a transformation to make things simpler.
Logarithm transforms products into sums."
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Let E = 2 log3 x + log3 5
We will use the logarethim properties to simplify the expression:
We know that: a*log b = log b^a
==> E = log3 (x^2) + log3 5
Now since the bases of the logarethim are the same we could use the properties of the product of two logarethims:
The, from logarethim properties we know that:
log a + log b = log a*b
==> E = log3 (x^2)*5
= log3 5x^2
Then we can wrtie the expression 2log3 x + log3 5 as follows:
==> 2log3 x + log3 5= log3 5x^2